A R K I V F O R M A T E M A T I K B a n d 4 n r 43

C o m m u n i c a t e d 6 D e c e m b e r 1961 b y O. FI~OSTMAlq

**A property o f bounded normal operators in Hilbert space **

B y GORAN BJORCK a n d YIDAR THOMI~E

Let H be a complex Hilbert space (i.e. a complete inner product space over
the complex numbers) with scalar product ( . , . ), norm I1" II, and identity ope-
rator I. For any bounded linear operator B in H we denote the spectrum b y
sp (B) and the point-spectrum by psp (B). The smallest circular disc containing
sp (B) is denoted by *dB, *the boundary of *dB *b y cB, and its center and radius b y
zB and *RB, *respectively. For a plane set S we denote the convex hull by cony (S).

For u and v in H we define

sin (u, v) = (1 - ](u, v)]2 ~ 89

### \

**Ilull~jlvll~!**(If u or v is zero, we define sin *(u,v)=O.) *

The purpose of this note is to prove and discuss the following
**Theorem **1. *Let N be a bounded normal operator in H. Then *

**sup/11Null2 ** **I(Nu'-~)12~ = R ~ , **

**=+o ~ Ilu[I 2 ** **Ilnll ~ ! ** (1)

or equivalently, SUPu.0 ~ sin (u, Nu) = RN. (2)

*Pro@ * We may evidently suppose t h a t RN> 0. For any complex ~t and a n y
*u E H * with I l u l l = l ,

**IlNull 2 - I ( N u , u)I s : II N ~ - ~ull 2 - I ( U u , **

**u) - ~ 12,****(3) **

Hence, taking 2 = zm

## sup/!lNull~ **I<~u-'~u)l~X = sup < **

**N u - z N u****2 - 1 ( N u , ~)-~NI:) **

=+o \ l l u l l ~ Ilull 4 ! u =1

**< sup **

**I I N u - z N u l l ~ = l l N - z N I l l 2 = R ~ ,****r **

Nu]]=l

since the norm and the spectral radius of the normal operator N - zN I are equal.

We will now exhibit a sequence

*unEH *

such t h a t
and

**Ilu.l[+l, ** **] **

**II Nu. - ** **u. II +R , ** **when **

**( N u . , u.) ~ zN **

n-+ ~ . (5)

The theorem will then be proved.

We first estabhsh t h a t zNE conv (sp (N)N cN). Otherwise, we could find a clo- sed half-circle

*h~CN *

having no point in common with sp (N). Since h and sp (N)
are compact, they would have a positive distance, and it would follow t h a t sp (N)
is contained in a smaller circular disc t h a n dN.
Thus there exist two or three distinct complex numbers ~t 1 .. . . . ~trE sp (N) such t h a t ( r = 2 or 3)

*12i - ZNr 12 = R2N' * *r * *} *

*ZN=~m~2~ *

with ~ m ~ = l and m l > 0 .
i = l i=1

(6)

r <0~+ ( i = l , r) i n H w i t h B y the properties of spectrum there exist sequences

*.tun fn:l *

^{. . . , }**liu )l I = l, ** **(7) **

**IINu2>-: o1,,= +11+0 when **

**n - - > ~ .**

**(8)**Since N is normal and the 2~ are distinct, (8) implies

(u~ >,u~))-->0 for i#?" when n - - > ~ . (9) Finally we define u, = Z [ : I m~ 89 u~ ). The verification of (5) is immediate, using (6),

(7), (8) and (9). This completes the proof.

The proof can, of course, also be carried through b y means of the spectral re- presentation of N (for notation, cf. [2]),

*N = f z E ( d x d y ) , * *z *

*= x + iy. *

I n this case the problem reduces to finding sup

*I(da), *

where
a n d ~/~ is the set of positive measures on sp (N) of the form

*da=(E(dxdy)u,u) *

with

**]lull **

= 1. We m a y also write
2I(da) =

*f f l z - ~ ] 2 da(z) da($). *

One of the authors ([1], Theorems 5 and 6) has studied the problem of maxi-

**ARKIV FOR MATEMATIK. B d 4 n r 4 3 **

mizing this integral for all positive measures on sp (N) with total mass 1, using essentially the above argument.

I t follows from the proof t h a t the sup in (1) and (2) is attained if and only
if we can find *u E H , I l u l l = l , * such t h a t

*[[ N u - ZN U 112 = f [ Z -- ZNI 2 (E(dxdy) u, u) = R~N, *

(Nu, u) = *f z(E(dxdy) u, u) = za. *

These equations express t h a t *( E ( d x d y ) u , u ) * has its s u p p o r t on cN and its center
of g r a v i t y in za. I n particular, the sup is attained if the points ~t in (6) and (8)
can be chosen as eigenvalues (and the sequences u~ ) as constant sequences of ei-
genvectors), i.e. if ZN E cony (psi) (N) n ca). This situation occurs for instance if
sp ( N ) = psp (N), which is, of course, always the case if H is finite-dimensional.

If N is self-adjoint, so t h a t sp (N) is a subset of the real line, then RN is half the diameter of the spectrum, and the sup in (1) and (2) can be attained only in the case discussed above.

I t is easy to construct an example of a (necessarily non self-adjoint) normal operator N in a (necessarily infinite-dimensional) Hilbert space H such t h a t psp (N) is e m p t y and still the sup in (1) and (2) is attained. (One can, for example, arrange t h a t sp (/V)=ca.)

We will a p p l y Theorem 1 and its proof to the following problem: Given
*u , / E H, * w h a t can be said a b o u t the spectrum of a n y bounded normal operator
with *N u = ] ? * The following result follows at once from (4).

Corollary 1. *Let u , / e H , * Ilull = 1, *and let r = *II/11 sin *(/,u). Let N be a bounded nor- *
*mal operator such that N u = / . Then *

o 2

r" ~< R a - I ( / , u) - ZN[ 2. (10)
*I n particular RN >~ r. *

Corollary 1 has the following consequence for self-adjoint operators:

Corollary 2. *Let u , / e H , I[ul] = l, r=[[/] I sin (/,u), and let (/,u) be real. Let A be a *
*sel]-ad]oint operator such that A u = ]. Then *conv(sp(A)) *contains some closed interval *
[(/, **u ) - ***:~, (/, ***u ) + flJ w i t h ~ ****= r ~ **

Corollary 2 is immediately obtained from Corollary 1. I n fact, conv(sp(A))

= [zA- RA, z~ +RA] and we have to prove t h a t

*(ZA + RA -- (/, U)) ((/, U) -- ZA § RA) >t r 2, *
which in this case is equivalent to (10).

As a trivial consequence of Corollary 2 we get:

Corollary 3. *Let u , / e H, *HuH = 1, r =

### II/Jl

sin*(/, u), and let (/, u) > O. Let A be a non-*

*negative, sel/-ad]oint operator with A u = / . Then*

s u p 2 / > (1, *u) + r2/(l, *^{u ) = }

### II 1113/(/,

*u).*

). G s p ( A )

This estimate, which can also be easily o b t a i n e d directly, is stronger t h a n t h e obvious ones, (/,u) a n d ]]/]]. I n fact, we h a v e b y the C a u c h y inequality,

**(/,u) **

**<**

**II/11 **

**<**

**II/llV(/,u). **

**II/llV(/,u).**

W e conclude this p a p e r b y discussing t h e following problem: G i v e n *u , / E H *
with I] u]l = 1, can we always find a n o r m a l o p e r a t o r N w i t h N u = / s u c h t h a t we
get e q u a l i t y in (10)? T h e answer is contained in the following

**Theorem 2. ****Let u, I E H , **

**Ilull **

^{= 1 , }

^{and let }^{r = }**II111 **

**sin ( l , u ) > 0 .**

**Assume that**the com-*plex number z and the positive number R satisly*

r3 = R2 - I(1, u) - zt 3. ( 1 1 )

*Then there exists a normal operator N with N u = / such that RN = R and ZN= Z. I n *
*particular, /or given u , / E H, I] u II = 1, it is always possible t o / l u g a normal operator N *
*such that N u = / and RN = r. I n this case, necessarily zN = z = *(/, u).

*Proo/. *W e first p r o v e the t h e o r e m for the case z = 0 a n d *(/,u) * real. Consider
t h e subspace G in H , s p a n n e d b y u a n d /. W e will p r o v e t h a t we can find t w o
orthogonal vectors Y~I a n d 92 such t h a t

I n fact, (12) is solved b y

**u=~01+~2' ** **1 ** **(12) **

/ = Ry~ 1 - RyJ2. j

Using the definition of r, we get f r o m (11)

T h e linear o p e r a t o r A defined b y

*A ( X l ~ l + x 2 ~ 3 § ) = x l R ~ l - x 2 R y ) 3 , * *v_L G, *

t h e n satisfies the conditions in T h e o r e m 2. T h e o p e r a t o r A is even self-adjoint.

T h e general case can now be reduced to the case considered above. P u t

**~/' **

**= e ' ~**

**zu),**where the real n u m b e r 0 is chosen such t h a t

**ARKIV FOR MATEMATIK. B d 4 n r 4 3 **

### (/',

**u ) = ego((/, u) - z)**

is real. W e e a s i l y f i n d (cf. (3)) t h a t

*r ' = II/']1 *sin ( u , l ' ) = Ilill sin *( u , / ) = r, *

a n d t h e r e f o r e r '2 = R 2 - (/', u) 2.

W e can t h u s f i n d a s e l f - a d j o i n t o p e r a t o r A w i t h c o n v ( s p ( A ) ) e q u a l t o t h e closed
i n t e r v a l [ - R, R] a n d such t h a t *A u *= ~'. T h e n o r m a l o p e r a t o r N = e -~ ~ A + z I t h e n
satisfies t h e c o n d i t i o n s in T h e o r e m 2.

T h e a u t h o r s a r e i n d e b t e d t o P r o f e s s o r L a r s H S r m a n d e r for s t i m u l a t i n g discus- sions on t h e c o n t e n t s of t h i s p a p e r .

*University o/Stockholm, Sweden. *

R E F E R E N C E S

1. BJORCK, G., Distributions of positive mass, which maximize a certain generalized energy integral. Ark. Mat. 3, 255-269 (1955).

2. RIESZ, F., and Sz.-NAGY, B., Legons d'analyse fonctionelle. Budapest, 1952.

Tryckt den 24 juli 1962

Uppsala 1962. Almqvist & Wiksells Boktryckeri AB